What is the length of an arc in terms of π that subtends an angle of `72^@` at the centre of a circle of radius 10 cm?

Answer

To find the length of an arc that subtends an angle of \(72^\circ\) at the center of a circle with a radius of \(10\) cm, we can use the formula for the length of an arc:

\[ L = R \times \theta \]

where:
- \(L\) is the length of the arc,
- \(R\) is the radius of the circle, and
- \(\theta\) is the angle in radians.

### Step 1: Convert the angle from degrees to radians
To convert degrees to radians, we use the conversion factor:

\[ \theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180} \]

For our problem:

\[ \theta = 72^\circ \times \frac{\pi}{180} \]

### Step 2: Simplify the angle in radians
Now, simplify the expression:

\[ \theta = 72 \times \frac{\pi}{180} = \frac{72\pi}{180} \]

We can simplify \(\frac{72}{180}\):

\[ \frac{72}{180} = \frac{2}{5} \]

Thus,

\[ \theta = \frac{2\pi}{5} \text{ radians} \]

### Step 3: Substitute the values into the arc length formula
Now we can substitute \(R = 10\) cm and \(\theta = \frac{2\pi}{5}\) into the arc length formula:

\[ L = R \times \theta = 10 \times \frac{2\pi}{5} \]

### Step 4: Calculate the length of the arc
Now, calculate \(L\):

\[ L = 10 \times \frac{2\pi}{5} = \frac{20\pi}{5} = 4\pi \text{ cm} \]

### Final Answer
The length of the arc is:

\[ \boxed{4\pi \text{ cm}} \]